isosctrlem1 - Metamath Proof Explorer

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Theorem isosctrlem1 26168

Description: Lemma for isosctr 26171. (Contributed by Saveliy Skresanov, 30-Dec-2016.)

**Assertion**
Ref	Expression
isosctrlem1	⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≠ π)

**Proof of Theorem isosctrlem1**
Step	Hyp	Ref	Expression
1		ax-1cn 11109	. . . . . . 7 ⊢ 1 ∈ ℂ
2		subcl 11400	. . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − 𝐴) ∈ ℂ)
3	1, 2	mpan 688	. . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 − 𝐴) ∈ ℂ)
4	3	adantr 481	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ∈ ℂ)
5		subeq0 11427	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴))
6	5	notbid 317	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (¬ (1 − 𝐴) = 0 ↔ ¬ 1 = 𝐴))
7	1, 6	mpan 688	. . . . . . 7 ⊢ (𝐴 ∈ ℂ → (¬ (1 − 𝐴) = 0 ↔ ¬ 1 = 𝐴))
8	7	biimpar 478	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → ¬ (1 − 𝐴) = 0)
9	8	neqned 2950	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ≠ 0)
10	4, 9	logcld 25926	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (log‘(1 − 𝐴)) ∈ ℂ)
11	10	imcld 15080	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℝ)
12	11	3adant2 1131	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℝ)
13	3	3ad2ant1 1133	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ∈ ℂ)
14	9	3adant2 1131	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ≠ 0)
15		releabs 15206	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ≤ (abs‘𝐴))
16	15	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (ℜ‘𝐴) ≤ (abs‘𝐴))
17		breq2 5109	. . . . . . . . . 10 ⊢ ((abs‘𝐴) = 1 → ((ℜ‘𝐴) ≤ (abs‘𝐴) ↔ (ℜ‘𝐴) ≤ 1))
18	17	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ((ℜ‘𝐴) ≤ (abs‘𝐴) ↔ (ℜ‘𝐴) ≤ 1))
19	16, 18	mpbid 231	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (ℜ‘𝐴) ≤ 1)
20		recl 14995	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ)
21	20	recnd 11183	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ)
22	21	subidd 11500	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((ℜ‘𝐴) − (ℜ‘𝐴)) = 0)
23	22	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≤ 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) = 0)
24		simpl 483	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≤ 1) → 𝐴 ∈ ℂ)
25	24	recld 15079	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≤ 1) → (ℜ‘𝐴) ∈ ℝ)
26		1red 11156	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≤ 1) → 1 ∈ ℝ)
27		simpr 485	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≤ 1) → (ℜ‘𝐴) ≤ 1)
28	25, 26, 25, 27	lesub1dd 11771	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≤ 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴)))
29	23, 28	eqbrtrrd 5129	. . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≤ 1) → 0 ≤ (1 − (ℜ‘𝐴)))
30	19, 29	syldan 591	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → 0 ≤ (1 − (ℜ‘𝐴)))
31		resub 15012	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (ℜ‘(1 − 𝐴)) = ((ℜ‘1) − (ℜ‘𝐴)))
32		re1 15039	. . . . . . . . . . 11 ⊢ (ℜ‘1) = 1
33	32	oveq1i 7367	. . . . . . . . . 10 ⊢ ((ℜ‘1) − (ℜ‘𝐴)) = (1 − (ℜ‘𝐴))
34	31, 33	eqtrdi 2792	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (ℜ‘(1 − 𝐴)) = (1 − (ℜ‘𝐴)))
35	1, 34	mpan 688	. . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (ℜ‘(1 − 𝐴)) = (1 − (ℜ‘𝐴)))
36	35	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (ℜ‘(1 − 𝐴)) = (1 − (ℜ‘𝐴)))
37	30, 36	breqtrrd 5133	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → 0 ≤ (ℜ‘(1 − 𝐴)))
38	37	3adant3 1132	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → 0 ≤ (ℜ‘(1 − 𝐴)))
39		neghalfpirx 25823	. . . . . 6 ⊢ -(π / 2) ∈ ℝ^*
40		halfpire 25821	. . . . . . 7 ⊢ (π / 2) ∈ ℝ
41	40	rexri 11213	. . . . . 6 ⊢ (π / 2) ∈ ℝ^*
42		argrege0 25966	. . . . . 6 ⊢ (((1 − 𝐴) ∈ ℂ ∧ (1 − 𝐴) ≠ 0 ∧ 0 ≤ (ℜ‘(1 − 𝐴))) → (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2)))
43		iccleub 13319	. . . . . 6 ⊢ ((-(π / 2) ∈ ℝ^* ∧ (π / 2) ∈ ℝ^* ∧ (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2))) → (ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2))
44	39, 41, 42, 43	mp3an12i 1465	. . . . 5 ⊢ (((1 − 𝐴) ∈ ℂ ∧ (1 − 𝐴) ≠ 0 ∧ 0 ≤ (ℜ‘(1 − 𝐴))) → (ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2))
45	13, 14, 38, 44	syl3anc 1371	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2))
46		pirp 25818	. . . . 5 ⊢ π ∈ ℝ⁺
47		rphalflt 12944	. . . . 5 ⊢ (π ∈ ℝ⁺ → (π / 2) < π)
48	46, 47	ax-mp 5	. . . 4 ⊢ (π / 2) < π
49	45, 48	jctir 521	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2) ∧ (π / 2) < π))
50		pire 25815	. . . . . . 7 ⊢ π ∈ ℝ
51	50	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → π ∈ ℝ)
52	51	rehalfcld 12400	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (π / 2) ∈ ℝ)
53		lelttr 11245	. . . . 5 ⊢ (((ℑ‘(log‘(1 − 𝐴))) ∈ ℝ ∧ (π / 2) ∈ ℝ ∧ π ∈ ℝ) → (((ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2) ∧ (π / 2) < π) → (ℑ‘(log‘(1 − 𝐴))) < π))
54	11, 52, 51, 53	syl3anc 1371	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (((ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2) ∧ (π / 2) < π) → (ℑ‘(log‘(1 − 𝐴))) < π))
55	54	3adant2 1131	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (((ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2) ∧ (π / 2) < π) → (ℑ‘(log‘(1 − 𝐴))) < π))
56	49, 55	mpd 15	. 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) < π)
57	12, 56	ltned 11291	1 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≠ π)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5105 ‘cfv 6496 (class class class)co 7357 ℂcc 11049 ℝcr 11050 0cc0 11051 1c1 11052 ℝ^*cxr 11188 < clt 11189 ≤ cle 11190 − cmin 11385 -cneg 11386 / cdiv 11812 2c2 12208 ℝ⁺crp 12915 [,]cicc 13267 ℜcre 14982 ℑcim 14983 abscabs 15119 πcpi 15949 logclog 25910

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-inf2 9577 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-ioc 13269 df-ico 13270 df-icc 13271 df-fz 13425 df-fzo 13568 df-fl 13697 df-mod 13775 df-seq 13907 df-exp 13968 df-fac 14174 df-bc 14203 df-hash 14231 df-shft 14952 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-rlim 15371 df-sum 15571 df-ef 15950 df-sin 15952 df-cos 15953 df-pi 15955 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-fbas 20793 df-fg 20794 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-ntr 22371 df-cls 22372 df-nei 22449 df-lp 22487 df-perf 22488 df-cn 22578 df-cnp 22579 df-haus 22666 df-tx 22913 df-hmeo 23106 df-fil 23197 df-fm 23289 df-flim 23290 df-flf 23291 df-xms 23673 df-ms 23674 df-tms 23675 df-cncf 24241 df-limc 25230 df-dv 25231 df-log 25912

This theorem is referenced by: isosctrlem2 26169

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